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  <h1 id="数学-高等数学 第3-4讲 一元函数微分学" class="content-subhead">数学-高等数学 第3-4讲 一元函数微分学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学 2 第3-4讲 一元函数微分学.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学 第3-4讲 一元函数微分学"></span>
  </p>
  <h2 id="3-4">第3-4讲 一元函数微分学</h2>
<h3 id="1">1. 导数定义</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f'(x_0) &= \lim_{\Delta x \to 0}\frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \\[3ex]
&= \lim_{x \to x_0}\frac{f(x) - f(x_0)}{x - x_0} \\
\end{split}\end{equation}
</script>
</p>
<p>无穷导数  <script type="math/tex"> \infty </script>  视为倒数不存在</p>
<blockquote class="content-quote">
<p>常用性质：<script type="math/tex"> f(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f'(x) </script> 为奇函数 <script type="math/tex"> \Rightarrow f''(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f^{(3)}(x) </script> 为奇函数&hellip;</p>
<p>极限存在 ⟺ <script type="math/tex">f_-'(x_0)=f_+'(x_0)</script>
</p>
</blockquote>
<h3 id="2">2. 微分定义</h3>
<p>设函数 <script type="math/tex">y=f(x)</script> 在点 <script type="math/tex">x_0</script> 的某领域内有定义，且 <script type="math/tex">x_0+\Delta x</script> 在该领域内，对于函数增量<br />
<script type="math/tex; mode=display">
\Delta y=f(x_0+\Delta x)-f(x_0)
</script>
<br />
若存在与 <script type="math/tex">\Delta x</script>
<strong>无关</strong>的，而仅与 <script type="math/tex">x</script>
<strong>有关</strong> 的常数 <script type="math/tex">A</script>，使得<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\Delta y &= A\Delta x+o(\Delta x) \\[1ex]
&= 线性增量 + 高阶无穷小量
\end{split}\end{equation}
</script>
<br />
则称 <script type="math/tex">f(x)</script> 在 <script type="math/tex">x_0</script> 处 <strong>可微</strong>，并称 <script type="math/tex">A\Delta x</script> 为 <script type="math/tex">f(x)</script> 在点 <script type="math/tex">x_0</script> 处的 <strong>微分</strong>，记作<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dy\bigg|_{x=x_0} &= A\Delta x \\[1ex]
&= Adx \\[1em]
导数：f'(x_0) = \cfrac{dy}{dx}\bigg|_{x=x_0} &= A
\end{split}\end{equation}
</script>
</p>
<h3 id="3">3. 导数与微分的计算</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
积的导数： [u(x)v(x)]' &= u'(x)v(x) + u(x)v'(x) \\[2ex]
积的微分： d[u(x)v(x)] &= du(x)v(x) + u(x)dv(x) \\[4ex]
商的导数： \bigg[\cfrac{u(x)}{v(x)}\bigg]' &= \cfrac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}, v(x) \neq 0 \\[2ex]
商的微分： d\bigg[\cfrac{u(x)}{v(x)}\bigg] &= \cfrac{du(x)v(x) - u(x)dv(x)}{[v(x)]^2}, v(x) \neq 0 \\[4ex]
复合函数的导数： \{f[g(x)]\}' &= f'[g(x)]g'(x) \\[2ex]
复合函数的微分： d\{f[g(x)]\} &= f'[g(x)]g'(x)dx \\[4ex]
\end{split}\end{equation}
</script>
</p>
<h3 id="4">4. 反函数求导</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y_x'&=\cfrac{1}{x_y'} \\[2ex]
推导过程：x_y'&=\cfrac{dx}{dy}=\cfrac{1}{\cfrac{dy}{dx}}=\cfrac{1}{y_x'} \\[1ex]
x_{yy}''&=\cfrac{d\cfrac{dx}{dy}}{dy}=\cfrac{d\cfrac{1}{y_x'}}{dy}=\cfrac{d\cfrac{1}{y_x'}}{dx}·\cfrac{1}{y_x'} \\[1ex]
&= -\cfrac{y_{xx}''}{(y_x')^2}·\cfrac{1}{y_x'} \\[1ex]
&=-\cfrac{y_{xx}''}{(y_x')^3}\\[2em]
y_x'&=\cfrac{1}{x_y'} \\
y_{xx}''&=-\ x_{yy}''·(y_x')^3=-\ \cfrac{x_{yy}''}{(x_y')^3}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>示例：求反函数 <script type="math/tex">y=\cfrac{1}{a}\arctan\cfrac{1}{a}x</script> 的导数，易得原函数 <script type="math/tex">y=a\tan ax</script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{1}{a}(\arctan\cfrac{1}{a}x)'
&=\cfrac{1}{(a\tan ay)'} \\[1ex]
&=\cfrac{1}{a(\cfrac{\sin ay}{\cos ay})'} \\[1ex]
&=\cfrac{1}{a(\cfrac{a\cos ay\cos ay+a\sin ay\sin ay}{\cos^2 ay})} \\[1ex]
&=\cfrac{\cos^2 ay}{a^2} \\[1ex]
&=\cfrac{\cos^2 ay}{a^2(\cos^2 ay + \sin^2 ay)} \\[1ex]
&=\cfrac{1}{a^2(1 + \tan^2 ay)} \\[1ex]
&=\cfrac{1}{a^2(1 + \cfrac{1}{a^2}x^2)} \\[1ex]
&=\cfrac{1}{a^2 + x^2} \\[1ex]
另外,\ \ \ \ \lim_{x\to0}\cfrac{\cfrac{\cos^2 ax}{a^2}}{\cfrac{1}{a^2+x^2}}&=(1+\cfrac{1}{a^2}x^2)\cos^2 ax=1 \\[1ex]
可见,\ \ \ \ \cfrac{cos^2 x}{a^2}\ 与&\ \cfrac{1}{a^2+x^2}\ 为等价无穷小
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h3 id="5">5. 幂指函数求导</h3>
<p>
<script type="math/tex; mode=display">
u(x)^{v(x)}=e^{v(x)\ln u(x)} \\[2em]
\{u(x)\gt0, u(x)\neq1\}
</script>
</p>
<h3 id="6">6. 参数方程求导</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dy}{dx} &= \cfrac{\cfrac{dy}{dt}}{\cfrac{dx}{dt}} = \cfrac{\psi'(t)}{\varphi'(t)} \\[2ex]
\cfrac{d^2y}{dx^2} &= \cfrac{d(\cfrac{dy}{dx})}{dx} = \cfrac{\cfrac{d(\cfrac{dy}{dx})}{dt}}{\cfrac{dx}{dt}}= \cfrac{\psi''(t)\varphi'(t)-\psi'(t)\varphi''(t)}{[\varphi'(t)]^3}
\end{split}\end{equation}
</script>
</p>
<h3 id="7">7. 莱布尼茨公式</h3>
<p>
<script type="math/tex; mode=display">
(uv)^{(n)} = u^{(n)}v + C^1_nu^{(n-1)}v' + \cdots + C^{n-1}_nu'v^{(n-1)} + uv^{(n)}
</script>
</p>
<h3 id="8">8. 可微、可导、连续、可积的关系</h3>
<ul>
<li>
<p>可微 ⟺ 可导 ⟹ 连续 ⟹ 可积</p>
</li>
<li>
<p>可导的条件：（左极限要等于右极限）</p>
<ul>
<li>如 <script type="math/tex"> y = |x|, \lim\limits_{x \to 0^-} = -1, \lim\limits_{x \to 0^+} = 1, 在\ x = 0\ 处不可导 </script>
</li>
</ul>
</li>
</ul>
</div>
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  <div class="navigation">
  <ul class="pure-menu-list">
    <li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#3-4">第3-4讲 一元函数微分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1">1. 导数定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2">2. 微分定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3">3. 导数与微分的计算</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4">4. 反函数求导</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5">5. 幂指函数求导</a>
</li>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6">6. 参数方程求导</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#7">7. 莱布尼茨公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#8">8. 可微、可导、连续、可积的关系</a>
</li>

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